Erdős Problem 728

Reference: erdosproblems.com/728

theorem Erdos728.erdos_728 : (∀ (ε C : ℝ), 0 < ε → 0 < C → ∃ (a : ℕ) (b : ℕ) (n : ℕ), ε * ↑n < ↑a ∧ ε * ↑n < ↑b ∧ a.factorial * b.factorial ∣ n.factorial * (a + b - n).factorial ∧ ↑a + ↑b > ↑n + C * Real.log ↑n) ↔ sorry

Let $\varepsilon, C > 0$. Are there integers $a, b, n$ such that $$a > \varepsilon n,\quad b > \varepsilon n, \quad a!\, b! \mid n!\, (a + b - n)!, $$ and $$ a + b > n + C \log n ?$$